Fine structure of the entanglement entropy in the O(2) model

We compare two calculations of the particle density in the superfluid phase of the O(2) model with a chemical potential $\mu$ in 1+1 dimensions. The first relies on exact blocking formulas from the Tensor Renormalization Group (TRG) formulation of the transfer matrix. The second is a worm algorithm. We show that the particle number distributions obtained with the two methods agree well. We use the TRG method to calculate the thermal entropy and the entanglement entropy. We describe the particle density, the two entropies and the topology of the world lines as we increase $\mu$ to go across the superfluid phase between the first two Mott insulating phases. For a sufficiently large temporal size, this process reveals an interesting fine structure: the average particle number and the winding number of most of the world lines in the Euclidean time direction increase by one unit at a time. At each step, the thermal entropy develops a peak and the entanglement entropy increases until we reach half-filling and then decreases in a way that approximately mirrors the ascent. This suggests an approximate fermionic picture.

Figure 1

Intensity plot for the thermal entropy of the O(2) model on a $4\times 128$ lattice in the $\beta -\mu$ plane. The dark (blue) regions are close to zero and the light (yellow ochre) regions peak near $ln2$. The MI phase with $\rho =0$ is below the lowest light band, the MI phase with $\rho =1$ is above the highest light band and there is a single SF phase in between the two MI phases.

Figure 2

Graphical representation of an allowed configuration of $\left\{n\right\}$ for a 4 by 32 lattice, $\beta =0.1$ and $\mu =3.0$. The uncovered links on the grid have $n=0$, the more pronounced dark lines have $\left|n\right|=1$, and the wider lines have $n=2$. The signs of the links with $\left|n\right|=1$ lines are discussed in Appendix pp1. The large dots (red) on the $t=0$ boundary need to be identified with the corresponding dots on the $t=32$ boundary in pairs with the same $x$ coordinate. Similarly, the slightly smaller dots (blue) on the $x=0$ and $x=4$ boundaries have to be identified in pairs with the same $t$ coordinate.

Figure 3

Graphical representation of the transfer matrix ${\mathbb{T}}_{(n_1,n_2,\cdots ,n_{L_x})(n_1^{\prime },n_2^{\prime },\cdots ,n_{L_x}^{\prime })}$ as defined in Eq. (5). The horizontal indices are summed over and the vertical indices are left open as explained in the text.

Figure 4

Graphical representation of the coarse graining truncation of the transfer matrix described in Fig. 3. The blocking procedure of replacing blocks of two sites in the transfer matrix into a single site represented by the Y-shaped structure.

Figure 5

The particle number distribution $P\left(n\right)$ with $\beta =0.1$ when $\mu$ varies between the MI and SF phases. For $\mu =2.80$ and 2.85, we have only $n=0$. For $\mu =2.90$, 2.95, and 3.0, there are three visible groups of bins. With $\mu$ increasing, the distribution shifts to the right with larger most probable particle number.

Figure 6

Comparison of the particle number distribution $P\left(n\right)$ between the worm algorithm and TRG with different $D_s$.

Figure 7

Illustration of the entanglement entropy calculation. The horizontal lines represent the traces over the space indices. There are $L_t$ of them, the missing ones being represented by dots. The two vertical lines represent the traces over the blocked time indices in $A$ and $B$.

Figure 8

Entanglement entropy (EE, dash line) and thermal entropy (TE, solid line) for $\beta =0.1, L_x=4$, and $L_t=16$, 32, 64, and 128.

Figure 9

The two lowest energy levels (EL, solid lines) as a function of $\mu$ for $L_x=4,L_t=256,\beta =0.1,\mathrm{and}{D}_s=101$. As $\mu$ increases, lines of successive slopes 0, $-1, -2, -3$, and $-4$ are at the lowest level. At each crossing, the thermal entropy(dash dot line) jumps. The values of thermal entropy are shifted vertically by $-0.5$ to make the figure readable.

Figure 10

Worm configurations for $\mu =2.93, n=1$ (a); $\mu =3.00, n=2$ (b); $\mu =3.07, n=3$ (c) and 3.14, $n=4$ (d). In all cases, $\beta =0.1, L_x=4$, and $L_t=256$.

Figure 11

Fine structure of the entanglement entropy (EE, blue circle), thermal entropy (TE, green square), and particle density $\rho$ (red triangle) spanning the $\mathrm{MI}(\rho =0$), SF, and $\mathrm{MI}(\rho =1$) phases successively with $\mu$ increasing at fixed $\beta =0.1$. There are three different system sizes: $L_x=4$ with $L_t=256, L_x=8$ with $L_t=512$, and $L_x=16$ with $L_t=1024$, respectively. The thermal entropy has $L_x$ peaks culminating near $ln2\simeq 0.69$; $\rho$ goes from 0 to 1 in $L_x$ steps, and the entanglement entropy has an approximate mirror symmetry near half filling where it peaks.

Figure 12

Energy levels for $\beta =0.1,L_x=4$.